##
**New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach.**
*(English)*
Zbl 1048.53026

The authors prove a new differential Harnack inequality of Li-Yau-Hamilton type (LYH inequality) for the Ricci flow. This inequality applies to solutions of the Ricci flow coupled to a one-form and a two-form solving Hodge-Laplacian heat-type equations.

Theorem. Let \(\{ M,g(t) \}\) be a solution of the Ricci flow on a closed manifold and a time interval \([0, \omega )\). Let \(A_0\) be a two-form which is closed at \(t=0\), and let \(E_0\) be a one-form which is closed at \(t=0\). Then there is a solution \(A(t)\) of \[ \frac{\partial}{\partial t} A = \Delta_d A, \quad A(0)=A_0, \] and a solution \(E(t)\) of \[ \frac{\partial}{\partial t} E = \Delta_d E - d| A|_g^2, \quad E(0)=E_0, \] which exist for \(t\in [0, \omega )\), where \(-\Delta_d=d\delta + \delta d\) is the Hodge-de Rham Laplacian. Suppose that the quadratic \[ \Psi(A,E,U,W) = Rm(U,U)-2\langle \nabla_WA,U\rangle +| A(W)|^2-\langle \nabla_WE,U\rangle = \]

\[ R_{ijkl}U^{ij}U^{kl} + 2 W^j\nabla_jA_{kl}U^{lk} + (g^{pq}A_{jp}A_{lq}-\nabla_jE_l)W^jW^l) \] is nonnegative at \(t=0\) for any two-form \(U\) and one-form \(V\). Then the matrix inequality \(\Psi (A,E,U,W)\geq 0\) persists for all \(t\in [0, \omega )\).

The obtained LYH inequality leads to the following trace inequalities.

Corollary 1. Under the hypotheses above, the trace inequality \[ 0\leq Rc(W,W) - 2 (\delta A) (W) +| A|^2 +\delta E \] persists for all \(t\in [0, \omega )\).

Corollary 2. Let \(\{ M,g(t) \}\) be a Kähler solution of the Ricci flow with nonnegative curvature operator on a closed manifold \(M\). Let \(A_0\) be a two-form which is closed at \(t=0\), and let \(E_0\) be a one-form which is closed at \(t=0\). Then for any two-form \(U\), one-form \(W\), and all \(t>0\) such that the solution exists, one has the matrix estimate \[ 0\leq Rm(U,U)-2\langle \nabla_W\rho,U\rangle +\frac{1}{4t^2}| W|^2 +\frac{1}{t}Rc(W,W) +Rc^2(W,W)+\frac{1}{2}(\nabla\nabla R)(W,W), \] where \(\rho\) is the Ricci form. By setting \(U=X\wedge W\) and tracing over \(W\), this implies the trace estimate \[ 0\leq \frac{\partial}{\partial t}R +\frac{2R}{t}+\frac{n}{2t^2} +2\langle \nabla R,X\rangle + 2Rc(X,X) \] for any one-form \(X\).

Corollary 3. Let \(\{ M,g(t) \}\) be a solution of the Ricci flow on a closed surface. If \((\phi, f)\) is a pair solving the system \[ \frac{\partial}{\partial t} \phi = \Delta \phi + R\phi, \quad \frac{\partial}{\partial t} f =\Delta f +\phi^2, \] then the trace inequality \[ 0\leq R| X|^2 + \langle \nabla \phi,X\rangle + \frac{\partial}{\partial t} f \] is preserved.

Theorem. Let \(\{ M,g(t) \}\) be a solution of the Ricci flow on a closed manifold and a time interval \([0, \omega )\). Let \(A_0\) be a two-form which is closed at \(t=0\), and let \(E_0\) be a one-form which is closed at \(t=0\). Then there is a solution \(A(t)\) of \[ \frac{\partial}{\partial t} A = \Delta_d A, \quad A(0)=A_0, \] and a solution \(E(t)\) of \[ \frac{\partial}{\partial t} E = \Delta_d E - d| A|_g^2, \quad E(0)=E_0, \] which exist for \(t\in [0, \omega )\), where \(-\Delta_d=d\delta + \delta d\) is the Hodge-de Rham Laplacian. Suppose that the quadratic \[ \Psi(A,E,U,W) = Rm(U,U)-2\langle \nabla_WA,U\rangle +| A(W)|^2-\langle \nabla_WE,U\rangle = \]

\[ R_{ijkl}U^{ij}U^{kl} + 2 W^j\nabla_jA_{kl}U^{lk} + (g^{pq}A_{jp}A_{lq}-\nabla_jE_l)W^jW^l) \] is nonnegative at \(t=0\) for any two-form \(U\) and one-form \(V\). Then the matrix inequality \(\Psi (A,E,U,W)\geq 0\) persists for all \(t\in [0, \omega )\).

The obtained LYH inequality leads to the following trace inequalities.

Corollary 1. Under the hypotheses above, the trace inequality \[ 0\leq Rc(W,W) - 2 (\delta A) (W) +| A|^2 +\delta E \] persists for all \(t\in [0, \omega )\).

Corollary 2. Let \(\{ M,g(t) \}\) be a Kähler solution of the Ricci flow with nonnegative curvature operator on a closed manifold \(M\). Let \(A_0\) be a two-form which is closed at \(t=0\), and let \(E_0\) be a one-form which is closed at \(t=0\). Then for any two-form \(U\), one-form \(W\), and all \(t>0\) such that the solution exists, one has the matrix estimate \[ 0\leq Rm(U,U)-2\langle \nabla_W\rho,U\rangle +\frac{1}{4t^2}| W|^2 +\frac{1}{t}Rc(W,W) +Rc^2(W,W)+\frac{1}{2}(\nabla\nabla R)(W,W), \] where \(\rho\) is the Ricci form. By setting \(U=X\wedge W\) and tracing over \(W\), this implies the trace estimate \[ 0\leq \frac{\partial}{\partial t}R +\frac{2R}{t}+\frac{n}{2t^2} +2\langle \nabla R,X\rangle + 2Rc(X,X) \] for any one-form \(X\).

Corollary 3. Let \(\{ M,g(t) \}\) be a solution of the Ricci flow on a closed surface. If \((\phi, f)\) is a pair solving the system \[ \frac{\partial}{\partial t} \phi = \Delta \phi + R\phi, \quad \frac{\partial}{\partial t} f =\Delta f +\phi^2, \] then the trace inequality \[ 0\leq R| X|^2 + \langle \nabla \phi,X\rangle + \frac{\partial}{\partial t} f \] is preserved.

Reviewer: Vasyl Gorkaviy (Kharkov)

### MSC:

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |